Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

Abstract

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

Reduced word manipulation: patterns and enumeration

“…Note that barred patterns can also be expressed in the more general language of mesh patterns, as introduced in [9].…”

Distributions of mesh patterns of short lengths

“…in (4) enumerating such A's. Further, since ab is an occurrence of p, to the right of a in π we must have a non-empty permutation, which can be any, and such permutations are counted by F (x) − 1.…”

“…Patterns in permutations and words have attracted much attention in the literature (see [8] and references therein), and this area of research continues to grow rapidly. The notion of a mesh pattern, generalizing several classes of patterns, was introduced by Brändén and Claesson [4] to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. A pair (τ, R), where τ is a permutation of length k and R is a subset of 0, k × 0, k , where 0, k denotes the interval of the integers from 0 to k, is a mesh pattern of length k. Let (i, j) denote the box whose corners have coordinates (i, j), (i, j + 1), (i + 1, j + 1), and (i + 1, j).…”

The Peak Statistics on Simsun Permutations

“…For example, (1,5,3,4)(2) is not a simsun permutation of the second kind since when we remove the letter 5, the resulting permutation (1,3,4)(2) contains a double excedance. Let SS n be the set of the simsun permutations of the second kind of length n. It is clear that exc (π) = cpk (π) for π ∈ SS n .…”